I'm curious to know if throughout my rebirths I have stayed the same religion, race, etc. or if I was of a different classification? I wonder this because Hinduism is an all-embracing religion and so it is not necessary for one to be Hindu to be graced and/or blessed by God? I am Hindu myself and would never change that, but I really do wonder if/how identities that are derived from social constructs traverse through all of the rebirths? I thought only our soul travels through rebirths.
I see two questions, here the answers for the two.
it is not necessary for one to be Hindu to be graced and/or blessed by God?
No. God's grace for you is not dependent on the variation in the form, but in the growth of intensity of your Bhakthi. Even within Hinduism, the journey from a Sakaara God, and a belief in many forms of Gods, usually tends to evolve towards one-pointed devotion to a single name and form (such as that of Arunagirinathar or Prahlada) and onwards towards an idea of God that transcends names and forms. The book by Sadhu Om, The Path of Sri Ramana (Part two) illustrates this journey well.
How identities that are derived from social constructs traverse through all of the rebirths :
In Hinduism, there are three planes of existence, the physical body is only of the gross plane, the other two being sukshma sarira and karana sarira, (subtle form and causal form). The key element that gets carried across births from the actions of a human life, are not the actions themselves, but the subtle impressions you gather from the actions. These are called Vasanas and Samskaras. A subset of these are almost the determinant of the starting point of the next birth, somewhat like the closing balance of your ledger for the current financial year, is the sum of all your transactions in the earlier years.
When you are writing a drama, if sleep comes in, you stop writing and retire to bed. As soon as you get up, you continue to write from where you have left the previous night. Even so, when you take up a new incarnation, you begin to continue the work which you had left unfinished in your previous life in accordance with the current of Vasanas of your past life.